Let p be prime. Let R be a discrete valuation ring of characteristic p with field of fractions K. Let C_p² denote the elementary abelian group of order p². In this paper we use Greither's approach for classifying short exact sequences of finite R-group schemes to classify R-Hopf orders H in the group ring KC_p², reproducing a result of Tossici. We then go further by providing an explicit description of the correspondence between these Hopf orders H and their duals H*, and also by explicitly describing their endomorphisms rings. Thus we are able to identify the Raynaud orders within this classification.